I have a series of character designs in Flash built in different symbols, eyes, head, etc. I need to export them to illustrator keeping them vector but when I use the FXG export the file fxg doesn't appear to 'remember' what frame the symbols were referencing and so the image isn't correct...

What exactly do you mean by it doesn't 'remember'? Need more information there.

However, please note that if there are objects on the stage that don't have corresponding tags recognized in the FXG format, they are exported as bitmaps files. The list of such unsupported content-types are bitmaps, some filters, some blend modes, gradients, masks, and 3D. You may want to see if your content has any of these and that's the reason you are facing issues. Also note that even if most of the content on stage can be exported to FXG, some of the other applications may not be able to read them.

Flash Pro CC 2014 (v14.0.0.110) is now available for download via the Creative Cloud App and Adobe website.

We have added SVG Export feature to Flash Pro with this new release. You will now be able to export out vector content from the selected frame as an SVG image that can be opened directly in a Browser and even imported in Adobe Illustrator.